History of Inverted-Pendulum Systems
Kent H. Lundberg
Franklin W. Olin College of Engineering, Needham, Mass. 02492
email: klund@alum.mit.edu
Taylor W. Barton
Massachusetts Institute of Technology, Cambridge, Mass. 02139
Abstract: The inverted-pendulum system is a favorite example system and lecture demonstration of students and educators in physics, dynamics, and control. This system is a simple
and valuable laboratory representation of an unstable mechanical system. This paper traces the
early history of the inverted-pendulum system, and also compares several of the early treatments
from the literature between 1960 and 1970.
1. INTRODUCTION
The inverted-pendulum system is a favorite example system and lecture demonstration of students and educators
in physics, dynamics, and control. This system is a simple
and valuable laboratory representation of an unstable mechanical system. It is also often used to model the control
problems encountered in the flight of rockets and missiles
in the initial stages of launch, when the airspeed is too
small for aerodynamic stability.
2. SOME HISTORY
Roberge (1960) demonstrated a solution to the invertedpendulum system at M.I.T. in his aptly named bachelor’s
thesis, “The Mechanical Seal”. Roberge’s research was
supervised by Leonard Gould.
Systems with multiple independent inverted pendula were
described by Higdon and Cannon (1963) at Stanford.
Higdon’s article acknowledges Roberge’s work and credits
Claude Shannon (father of information theory and avid
unicyclist) with suggesting the multiple-inverted-pendula
mechanical system in a prominent footnote:
This model was suggested to the second author
by Prof. Claude Shannon, of MIT. (Experiments with a single pendulum are reported in
an SB thesis entitled “The Mechanical Seal”
by Roberge at MIT, May 1960.)
Schaefer and Cannon (1966) discussed jointed and flexible
inverted-pendulum systems. This article (which shares a
coauthor with Higdon and Cannon (1963)) also credits
Shannon with suggesting the system, but does not mention
Roberge’s work.
Truxal (1965) wrote a set of lecture notes on state-space
models and control using the dual-inverted-pendulum system as an example. By the end of the 1960s, discussions
of the single inverted-pendulum system were included in
popular textbooks such as Cannon (1967), Dorf (1967),
and Ogata (1970) (which all reference Higdon and Cannon
(1963) in their discussions of the inverted pendulum).
Stabilization of a pendulum in the inverted configuration
by a vertically oscillating base is a favorite example of
classes in physics and classical mechanics. This example
system was treated by a series of articles in American
Journal of Physics by Phelps and Hunter (1965), Blitzer
(1965), and Kalmus (1970).
A recent article by Åström and Furuta (2000) claims that
“Inverted pendulums have been classic tools in the control
laboratories since the 1950s,” but their earliest citation is
Schaefer and Cannon (1966).
Swing-up control of an inverted-pendulum system was
demonstrated by Mori et al. (1976) (which cites Schaefer and Cannon (1966) but nothing earlier). The rotary
pendulum system was suggested as an alternative to the
cart-on-track system by Furuta et al. (1991).
3. MODELS
The analysis of the inverted-pendulum system and the
design of the stabilizing controls by various authors shows
interesting differences. Stabilizing the angle of the pendulum is straightforward, but a significant (and often
overlooked) difficulty exists in maintaining small deviations in cart travel. Without position control, in addition
to angle control, the cart will quickly run out of track.
This section compares the early treatments by Roberge
(1960), Higdon and Cannon (1963), Cannon (1967), Dorf
(1967), and Ogata (1970). In addition, this section reviews
the treatment by Siebert (1986), which is one of the few
complete discussions of the cart-position problem in the
textbook literature.
3.1 Roberge (1960)
Summing forces at the head of the broom (pendulum)
Roberge (1960) finds the transfer function
s2 /g
Θ
(s) =
X
(L/g)s2 − 1
where L is the length of the pendulum and g is the
acceleration of gravity.
Bode Diagram
Gm = −6.25 dB (at 1.43 rad/sec) , Pm = −7.04 deg (at 0.771 rad/sec)
Bode Diagram
Gm = 9.79 dB (at 16.8 rad/sec) , Pm = 20.7 deg (at 7.47 rad/sec)
50
100
0
Magnitude (dB)
Magnitude (dB)
50
0
−50
−50
−100
−100
−150
−135
−150
−135
−180
Phase (deg)
Phase (deg)
−180
−225
−225
−270
−270
−315
−315
−360
−3
10
−360
−3
−2
10
−1
10
0
10
1
10
Frequency (rad/sec)
2
10
−2
−1
10
10
3
10
10
Fig. 1. Bode plot of the compensated loop transfer function, showing 21 degress of phase margin. Reproduced
in MATLAB from the transfer functions in Roberge
(1960).
0
10
Frequency (rad/sec)
1
2
10
3
10
10
Fig. 3. Bode plot of the loop transfer function, including
the effects of the position-compensation loop, showing
18 degress of phase margin at the critical cross-over
frequency. Reproduced in MATLAB from the transfer
functions in Roberge (1960).
Nyquist Diagram
1
Nyquist Diagram
4
0.8
0.6
3
0.4
Imaginary Axis
2
Imaginary Axis
1
0
0.2
0
−0.2
−0.4
−1
−0.6
−2
−0.8
−3
−1
−3
−4
−7
−6
−5
−4
−3
Real Axis
−2
−1
0
−2.5
−2
−1.5
−1
Real Axis
−0.5
0
0.5
1
1
Fig. 2. Nyquist plot of the compensated loop transfer
function, showing a negative encirclement of the −1
point. Since the system has one open-loop pole in
the right-half plane, the closed-loop system is stable.
Reproduced in MATLAB from the transfer functions
in Roberge (1960).
Fig. 4. Nyquist plot of the loop transfer function, including
the effects of the position-compensation loop, showing
two negative encirclements of the −1 point. Since the
system has two open-loop poles in the right-half plane
(one from the pendulum and one from the positivefeedback-integrator loop), the closed-loop system is
stable. Reproduced in MATLAB from the transfer
functions in Roberge (1960).
To stabilize the system, a compensator with transfer
function
K
(αds + 1)
G(s) = 2
·
2
s (cs + 1)
(ds + 1)
discusses a control scheme without position compensation.
Then, admitting that “it is desired to have the cart return
to a given position on the floor after correcting a given
initial disturbance,” a controller that stabilizes the cart
position is found.
is used, as shown in Figure 1, where the integrations
are implemented using electromechanical motor-tach units
(that is, true integrations). The (cs + 1) term in the
denominator models the lag of these motor-tach units.
The second term of the compensator transfer function
implements lead compensation to offset the integrator lag
and to push the closed-loop poles into the left-half plane.
The Nyquist criterion is used to illustrate the stability of
the system, as shown in Figure 2. The mechanical system
was built using a plotting table and electromechanical
integrators from the M.I.T. Dynamic Analysis and Control
Laboratory (described by Hall (1950)).
Two bang-bang controllers are synthesized for the system,
one with linear switching and one with a limiting nonlinearity. For the linear-switching case, the control law is
p
fẋ
u = a sign θ + θ̇ ρ2 /gr + ẋb − xk
Mg
Turning to the position of the cart, Roberge notes
Since the major loop as developed thus far
has a double integrator in the forward gain
path and no position feedback, drift becomes
a problem. Drift could cause the platform to
reach the limits of travel of [the cart] and
thus control would be lost. Even if no drift
is assumed in the loop, an initial synchro
misalignment with respect to vertical of only
one second of arc (certainly much smaller than
can be achieved in practice) would cause the
broom to reach the limits of travel in about 100
seconds. To eliminate this problem position
feedback was employed. . . The position signal
is summed with the synchro signal to form the
input of the first integrator. Polarity is chosen
to cause positive feedback—if the base of the
broom moves to the right, the synchro null
is effectively shifted towards the center of the
table, thus causing the broom base to move
slightly futher to the right, and the broom
handle tips inward. The net result is to force
the broom to fall back towards the center of
the table.
The result of this additional feedback loop is shown in the
Bode plot in Figure 3 and the Nyquist plot in Figure 4.
3.2 Higdon and Cannon (1963)
Higdon and Cannon (1963) find the linearized equations
of motion to be
mρ2 θ̈ = mrgθ − mrẍ
M ẍ = −mrθ̈ + fẋ ẋ + fv v
where θ is the pendulum angle, x is the position of the
cart, m is the pendulum mass, ρ is the pendulum radius
of gyration about the hinge line, r is the distance from
the hinge line to the pendulum center of mass, g is the
acceleration of gravity, M is the total system effective
mass, fẋ is the motor damping coefficient, fv is the voltage
force coefficient, and v is the applied voltage to the motor.
After recasting the equations of motion in normal coordinates and examining the resulting eigenvalues, Higdon first
where
p
gr/ρ2
p
.
b=
fẋ /M g − gr/ρ2
fẋ /M g
Higdon observes
It is interesting to note that for a damped
motor fẋ is negative, hence the coefficients
of x and ẋ are positive, indicating a positive
feedback loop around cart position. This result
was found by linear analysis also, but only after
considerable head scratching.
3.3 Cannon (1967)
In his textbook, Cannon (1967) finds the equations of
motion of the pendulum-and-cart system to be
(mC + m)ẍ + mlθ̈ = f
mlẍ + (J + ml2 )θ̈ − mglθ = 0
and the transfer function from force to angle
Θ
−ρ1 /lm
(s) = 2
F
s − σo2
where ρ1 = 3/(1 + 4mC /m) and
s
3(1 + mC /m)g
.
σo =
(1 + 4mC /m)l
The closed-loop system is stabilized using a lead compensator, and illustrated using the root-locus method, as
shown in Figure 5.
Cannon observes “The behavior of coordinate x (cart
position) during the controlled recovery is also of interest”,
but then leaves the details as an exercise for the reader:
Prob. 22.33 Design a simple auxiliary loop,
to be added to the stick-balancing system. . .
whose purpose is simply to control cart position x to be zero with a leisurely speed of
response (i.e., merely to keep the cart on the
premises) . . . The effectiveness of this control
may be demonstrated merely by (i) showing
that its characteristic equation has all stable
roots, and (ii) writing the overall system response function for an initial x, then using
FVT to show that x(∞) is 0. (iii) As an additional feature, IVT may be used (shrewdly) to
show that the initial velocity ẋ will be negative.
That is, the cart corrects an x error by first
backing up. Explain physically.
3.6 Siebert (1986)
Root Locus
15
Siebert (1986) contains one of the few complete discussions
of the cart-position problem in the textbook literature. He
starts with the equation of motion
10
mgl sin θ − mlẍ cos θ = I θ̈
Imaginary Axis
5
to develop the small-angle linearized transfer function from
cart position to angle
Θ
−mls2
H(s) = (s) = 2
.
X
Is − mgl
0
−5
−10
−15
−15
−10
−5
0
Real Axis
5
10
15
Fig. 5. Root-locus plot of the pendulum transfer function,
stabilized with lead compensation. Reproduced in
MATLAB from Figure 22.8 in Cannon (1967).
3.4 Dorf (1967)
Dorf (1967) finds the equations of motion (assuming the
cart M is much more massive than the pendulum m and
ignoring the moment of inertia) to be
M ÿ + mlθ̈ = u(t)
mlÿ + ml2 θ̈ − mglθ = 0
Using these equations to obtain the necessary first-order
differential equations, he finds the system matrix


01 0 0
mg
0 0 −
0

M 
A=
.
0 0 0 1
g
0
00
l
After suggesting sensors that could be used for full-state
feedback (potentiometers and tachometers on angle and
position), the system matrix is reduced (by throwing away
the states associated with cart position and velocity!) and
a stabilizing controller is found with
u(t) = h1 θ + h2 θ̇
where it is shown that for stability, h2 > 0 and h1 > g.
Unfortunately, this control scheme does not stabilize the
position of the cart.
3.5 Ogata (1970)
Ogata (1970) finds the same equations of motion as
Cannon
(J + ml2 )θ̈ + mlÿ − mglθ = 0
mlθ̈ + (mC + m)ẍ = u
and uses the same stabilizing control as Dorf,
u = M (aθ + bθ̇).
For stability, it is necessary that b > 0 and a > (1 +
m/M )g. Again, this solution does not control the position
of the cart.
Assuming the cart is driven by a motor with transfer
function
X
km
M (s) = (s) =
V
s(s + α)
and proportional-plus-integral compensator
a
,
K(s) = K 1 +
s
the system as shown in Figure 6 (with the desired pendulum angle Θ0 as the input variable and the actual
pendulum angle Θ as the output variable)
Θ
−M (s)H(s)
(s) =
Θ0
1 − K(s)M (s)H(s)
is shown to be stable, using the closed-loop transfer
function and the Routh criterion. However, it is noted that
the closed-loop transfer function to cart position has a pole
at the origin
−M (s)
X
(s) =
Θ0
1 − K(s)M (s)H(s)
as shown in Figure 7. Thus,
a succession of random disturbances will induce a “random walk” in the car’s position that
will sooner or later cause it to go off one end
or the other of the track. This can be avoided
by adding still another feedback path. . .
This additional feedback path is shown as positive feedback from cart position to pendulum angle
which implies that deviations of x(t) from its
zero position will induce motor inputs in a
direction that makes the error worse. But a
little reflection on how you move your hand
balancing a pointer will make it clear that
this counterintuitive result is indeed correct.
To achieve an ultimate motion of your hand
to the right, you must first move it sharply to
the left, displacing the pendulum angle to the
right so that you can then steadily move your
hand to the right under the pendulum.
This intuitive explanation is satisfying for students and
educators.
4. ADVANCES IN CONTROL EDUCATION 2009
The Eighth IFAC Symposium on Advances in Control
Education is meeting in Kumamoto, Japan in October
Θ0
+
M (s)
X
H(s)
Θ
+
K(s)
Fig. 6. Block diagram from reference angle Θ0 to pendulum angle Θ. Adapted from Figure 6.4-3 of Siebert (1986).
ml
−km s2 − g
X
I
.
(s) = ml
ml
Θ0
(Kkm − g)s +
(Kkm a − gα)
s s3 + αs2 +
I
I
Fig. 7. Closed-loop transfer function from reference angle to cart position showing the unstable pole at the origin.
Reproduced from Siebert (1986).
2009. As described in the Call for Papers (IFAC (2009)),
the theme of the conference is
Inverted pendulum has been utilized for evaluating all kinds of control algorithms developed
in control research field since its first success
by Prof. Furuta in 1975. Today, inverted pendulum is used as the best benchmark in laboratory. Control engineering education with inverted pendulum will be specifically addressed
by the ACE2009 program, where all kinds of
algorithms for inverted pendulum will be proposed and competed by practical experiments
in the site.
As shown in this paper, this history is incomplete.
REFERENCES
K. J. Åström and K. Furuta. Swinging up a pendulum by
energy control. Automatica, 36:287–295, 2000.
Taylor W. Barton. Stabilizing the dual inverted pendulum.
In Advances in Control Education, Kumamoto, Japan,
October 2009.
Leon Blitzer. Inverted pendulum. American Journal of
Physics, 33(12):1076–1078, December 1965.
Robert H. Cannon. Dynamics of Physical Systems, pages
703–710. McGraw-Hill, New York, 1967.
Richard C. Dorf. Modern Control Systems, pages 276–279.
Addison-Wesley, Reading, 1967.
Katsuhisa Furuta, Masaki Yamakita, and Seiichi
Kobayashi. Swing up control of inverted pendulum.
In Proceedings, International Conference on Industrial
Electronics, Control and Instrumentation, volume 3,
pages 2193–2198, Kobe, Japan, 1991.
Albert C. Hall. A generalized analogue computer for flight
simulation. Transactions of the AIEE, 69:308–320, 1950.
Donald T. Higdon and Robert H. Cannon. On the
control of unstable multiple-output mechanical systems.
ASME Publication 63-WA-148, American Society of
Mechanical Engineers, New York, 1963.
IFAC. Eighth IFAC Symposium on Advances in Control
Education, Call for Papers, October 2009. URL http://
ace2009.cs.kumamoto-u.ac.jp/CFP_ACE2009A.pdf.
Henry P. Kalmus. The inverted pendulum. American
Journal of Physics, 38(7):874–878, July 1970.
Kent H. Lundberg and James K. Roberge. Classical dualinverted-pendulum control. In Proceedings of the IEEE
Conference on Decision and Control, pages 4399–4404,
Maui, December 2003.
Shozo Mori, Hiroyoshi Nishihara, and Katsuhisa Furuta.
Control of unstable mechanical systems: Control of
pendulum. International Journal of Control, 23(5):673–
692, May 1976.
Katsuhiko Ogata. Modern Control Engineering, pages
277–279. Prentice-Hall, Englewood Cliffs, 1970.
F. M. Phelps and J. H. Hunter. An analytical solution of
the inverted pendulum. American Journal of Physics,
33(4):285–295, April 1965.
James K. Roberge. The mechanical seal. Bachelor’s thesis,
Massachusetts Institute of Technology, Cambridge, May
1960.
James K. Roberge. Propagation of the race (of analog
circuit designers). In Jim Williams, editor, Analog Circuit Design: Art, Science, and Personalities, chapter 10,
pages 79–87. Butterworth-Heinemann, Boston, 1991.
J. F. Schaefer and R. H. Cannon. On the control of
unstable mechanical systems. In Proceedings of the
1966 International Federation of Automatic Control,
volume 1, pages 6C.1–6C.13, London, 1966.
William McC. Siebert. Circuits, Signals, and Systems,
pages 177–182. MIT Press, Cambridge, 1986.
J. G. Truxal. State models, transfer functions, and simulation. Monograph 8, Discrete Systems Concept Project,
1965.
ABOUT THE AUTHORS
Lundberg and Roberge (2003) described the intuitive
construction of classical controllers for single- and dualinverted-pendulum systems, based on the intuitive singleinverted-pendulum controller of Siebert (1986) and the
intuitive dual-inverted-pendulum controller as first described by Roberge (1991). This intuitive dual-invertedpendulum control system was built and demonstrated by
Barton (2009).